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Transition-metal dichalcogenides (TMDs) are layered compounds that support many electronic phases, including various charge density waves, superconducting, and Mott insulating states. Their intercalation with magnetic ions introduces magnetic sublayers, which strongly influence the coupling between host layers, and feature various magnetic states adjustable by external means. Co$_{1/3}$NbS$_2$ hosts a particularly sensitive magnetic subsystem with the lowest magnetic ordering temperature in the family of magnetically intercalated TMDs, and the only one where the complete suppression of magnetic order under pressure has been recently suggested. By combining the results of several experimental methods, electronic ab initio calculations, and modeling, we develop insights into the mechanisms of electric transport, magnetic ordering, and their interaction in this compound. The elastic neutron scattering is used to directly follow the evolution of the antiferromagnetic order parameter with pressure and temperature. Our results unambiguously disclose the complete suppression of the observed magnetic order around 1.7 GPa. We delve into possible mechanisms of magnetic order suppression under pressure, highlighting the role of magnetic frustrations indicated by magnetic susceptibility measurements and ab-initio calculations. Electronic conduction anisotropy is measured in the wide temperature and pressure range. Here we show that the transport in directions along and perpendicular to layers respond differently to the appearance of magnetic ordering or the application of the hydrostatic pressure. We propose the 'spin-valve' mechanism where the intercalated Co ions act as spin-selective electrical transport bridges between host layers. The mechanism applies to various magnetic states and can be extended to other magnetically intercalated TMDs., Comment: 36 pages, 19 figures